Stochastic energy-exchange models of non-equilibrium. Cristian - PowerPoint PPT Presentation

SU ( 1 , 1 ) structure Discrete representation � � ξ i + 1 K + i | ξ i � = | ξ i + 1 � 2 K − i | ξ i � = ξ i | ξ i − 1 � � ξ i + 1 � K o i | ξ i � = | ξ i � 4 Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Discrete representation � � ξ i + 1 K + i | ξ i � = | ξ i + 1 � 2 K − i | ξ i � = ξ i | ξ i − 1 � � ξ i + 1 � K o i | ξ i � = | ξ i � 4 In a canonical base 1   0 0 1 0     4 ... ...  ...         1    5 2       2 4       K + K 0   = K − = i =     ... ... ... ... i   i       3     9    2    4         ...  ... ...    ...   Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Discete representation � � ξ i + 1 K + i | ξ i � = | ξ i + 1 � 2 K − i | ξ i � = ξ i | ξ i − 1 � � ξ i + 1 � K o i | ξ i � = | ξ i � 4 In this representation L f ( ξ ) = L SIP f ( ξ ) � ξ j + 1 � � ξ i + 1 � � [ f ( ξ i , j ) − f ( ξ )] + ξ j [ f ( ξ j , i ) − f ( ξ )] = ξ i 2 2 ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

SU ( 1 , 1 ) structure Intertwiner K + i D i ( · , ξ i )( x i ) = K + i D i ( x i , · )( ξ i ) K − i D i ( · , ξ i )( x i ) = K − i D i ( x i , · )( ξ i ) K o i D i ( · , ξ i )( x i ) = K o i D i ( x i , · )( ξ i ) From the creation operators x 2 � ξ i + 1 � i 2 D i ( x i , ξ i ) = D ( x , ξ i + 1 ) 2 Therefore x 2 ξ i i D i ( x i , ξ i ) = ( 2 ξ i − 1 )!! D i ( x i , 0 ) Cristian Giardin` a (UniMoRe)

Self-duality Cristian Giardin` a (UniMoRe)

Markov chain with finite state space Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Indeed Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Indeed LD ( · , ξ )( η ) = LD ( η, · )( ξ ) Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Indeed � L ( η, η ′ ) D ( η ′ , ξ ) = LD ( · , ξ )( η ) = LD ( η, · )( ξ ) η ′ Cristian Giardin` a (UniMoRe)

Markov chain with finite state space 1. Matrix formulation of self-duality ( L dual = L ) LD = DL T Indeed � L ( η, η ′ ) D ( η ′ , ξ ) = LD ( · , ξ )( η ) = LD ( η, · )( ξ ) = � L ( ξ, ξ ′ ) D ( η, ξ ′ ) η ′ ξ ′ Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed Ld ( η, ξ ) = dL T ( η, ξ ) Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed L ( η, ξ ) = Ld ( η, ξ ) = dL T ( η, ξ ) µ ( ξ ) Cristian Giardin` a (UniMoRe)

Self-Duality 2. trivial self-duality ⇐ ⇒ reversible measure µ 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed L ( η, ξ ) = Ld ( η, ξ ) = dL T ( η, ξ ) = L ( ξ, η ) µ ( ξ ) µ ( η ) Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdL T Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdL T = DL T Cristian Giardin` a (UniMoRe)

Self-Duality 3. S : symmetry of the generator, i.e. [ L , S ] = 0, d : trivial self-duality function, − → D = Sd self-duality function. Indeed LD = LSd = SLd = SdL T = DL T Self-duality is related to the action of a symmetry. Cristian Giardin` a (UniMoRe)

Self-duality of the SIP process Theorem 2 The process with generator L SIP is self-dual on functions � 1 � Γ η i ! � 2 D ( η, ξ ) = � 1 ( η i − ξ i )! � Γ 2 + ξ i i ∈ V Cristian Giardin` a (UniMoRe)

Self-duality of the SIP process Theorem 2 The process with generator L SIP is self-dual on functions � 1 � Γ η i ! � 2 D ( η, ξ ) = � 1 ( η i − ξ i )! � Γ 2 + ξ i i ∈ V Proof: [ L SIP , � K o i ] = [ L SIP , � K + i ] = [ L SIP , � K − i ] = 0 i i i i K + � Self-duality fct related to the simmetry S = e i Cristian Giardin` a (UniMoRe)

Boundary driven systems. Cristian Giardin` a (UniMoRe)

Brownian Momentum Process with reservoirs 2 2 x x i j BMP L     2 2     res res L T x L T x     R R N 2 L L 1 2 x x x x N N 1 1 Cristian Giardin` a (UniMoRe)

Inclusion Process with absorbing reservoirs j i SIP L         abs L       L abs 1 , 0 f 2 f f N 1 1 Cristian Giardin` a (UniMoRe)

Duality between BMP with reservoirs and SIP with absorbing boundaries Configurations ¯ ξ = ( ξ 0 , ξ 1 , . . . , ξ N , ξ N + 1 ) ∈ Ω dual = N N + 2 Cristian Giardin` a (UniMoRe)

Duality between BMP with reservoirs and SIP with absorbing boundaries Configurations ¯ ξ = ( ξ 0 , ξ 1 , . . . , ξ N , ξ N + 1 ) ∈ Ω dual = N N + 2 Theorem 3 The process { x ( t ) } t ≥ 0 with generator L BMP , res is dual to the process ξ ( t ) } t ≥ 0 with generator L SIP , abs on { ¯ � N x 2 ξ i � T ξ N + 1 D ( x , ¯ ξ ) = T ξ 0 � i L R ( 2 ξ i − 1 )!! i = 1 Cristian Giardin` a (UniMoRe)

CONSEQUENCES OF DUALITY From continuous to discrete: Interacting diffusions (BMP) studied via particle systems (SIP). From many to few: n -points correlation functions of N particles using n dual walkers Remark: n ≪ N From reservoirs to absorbing boundaries: Stationary state of dual process described by absorption probabilities at the boundaries Cristian Giardin` a (UniMoRe)

Proposition ξ ( a , b ) = P ( ξ 0 ( ∞ ) = a , ξ N + 1 ( ∞ ) = b | ξ ( 0 ) = ¯ ξ ) . Then Let P ¯ E ( D ( x , ¯ � T a L T b ξ )) = R P ¯ ξ ( a , b ) a , b Cristian Giardin` a (UniMoRe)

Proposition ξ ( a , b ) = P ( ξ 0 ( ∞ ) = a , ξ N + 1 ( ∞ ) = b | ξ ( 0 ) = ¯ ξ ) . Then Let P ¯ E ( D ( x , ¯ � T a L T b ξ )) = R P ¯ ξ ( a , b ) a , b Proof: E ( D ( x , ¯ t →∞ E x 0 ( D ( x t , ¯ ξ )) = lim ξ )) Cristian Giardin` a (UniMoRe)

Proposition ξ ( a , b ) = P ( ξ 0 ( ∞ ) = a , ξ N + 1 ( ∞ ) = b | ξ ( 0 ) = ¯ ξ ) . Then Let P ¯ E ( D ( x , ¯ � T a L T b ξ )) = R P ¯ ξ ( a , b ) a , b Proof: E ( D ( x , ¯ t →∞ E x 0 ( D ( x t , ¯ ξ )) = lim ξ )) ξ ( D ( x 0 , ¯ = lim ξ t )) t →∞ E ¯ Cristian Giardin` a (UniMoRe)

Proposition ξ ( a , b ) = P ( ξ 0 ( ∞ ) = a , ξ N + 1 ( ∞ ) = b | ξ ( 0 ) = ¯ ξ ) . Then Let P ¯ E ( D ( x , ¯ � T a L T b ξ )) = R P ¯ ξ ( a , b ) a , b Proof: E ( D ( x , ¯ t →∞ E x 0 ( D ( x t , ¯ ξ )) = lim ξ )) ξ ( D ( x 0 , ¯ = lim ξ t )) t →∞ E ¯ � N x 2 ξ i � T ξ N + 1 D ( x , ¯ ξ ) = T ξ 0 � i using L R ( 2 ξ i − 1 )!! i = 1 Cristian Giardin` a (UniMoRe)

Proposition ξ ( a , b ) = P ( ξ 0 ( ∞ ) = a , ξ N + 1 ( ∞ ) = b | ξ ( 0 ) = ¯ ξ ) . Then Let P ¯ E ( D ( x , ¯ � T a L T b ξ )) = R P ¯ ξ ( a , b ) a , b Proof: E ( D ( x , ¯ t →∞ E x 0 ( D ( x t , ¯ ξ )) = lim ξ )) ξ ( D ( x 0 , ¯ = lim ξ t )) t →∞ E ¯ � N x 2 ξ i � T ξ N + 1 D ( x , ¯ ξ ) = T ξ 0 � i using L R ( 2 ξ i − 1 )!! i = 1 ξ ( T ξ 0 ( ∞ ) T ξ N + 1 ( ∞ ) = E ¯ ) L R Cristian Giardin` a (UniMoRe)

Temperature profile ξ = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ⇒ D ( x , � � ξ ) = x 2 i site i ր ⇒ 1 SIP walker ( X t ) t ≥ 0 with X 0 = i Cristian Giardin` a (UniMoRe)

Temperature profile ξ = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ⇒ D ( x , � � ξ ) = x 2 i site i ր ⇒ 1 SIP walker ( X t ) t ≥ 0 with X 0 = i � � x 2 = T L P i ( X ∞ = 0 ) + T R P i ( X ∞ = N + 1 ) E i Cristian Giardin` a (UniMoRe)

Temperature profile ξ = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ⇒ D ( x , � � ξ ) = x 2 i site i ր ⇒ 1 SIP walker ( X t ) t ≥ 0 with X 0 = i � � x 2 = T L P i ( X ∞ = 0 ) + T R P i ( X ∞ = N + 1 ) E i � T R − T L � E ( x 2 i ) = T L + i N + 1 i ) = T R − T L � J � = E ( x 2 i + 1 ) − E ( x 2 Fourier ′ s law N + 1 Cristian Giardin` a (UniMoRe)

Energy covariance If � D ( x , � ξ ) = x 2 i x 2 ξ = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ⇒ j site i ր site j ր Cristian Giardin` a (UniMoRe)

Energy covariance If � D ( x , � ξ ) = x 2 i x 2 ξ = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 , 1 , 0 , . . . , 0 ) ⇒ j site i ր site j ր In the dual process we initialize two SIP walkers ( X t , Y t ) t ≥ 0 with ( X 0 , Y 0 ) = ( i , j ) Cristian Giardin` a (UniMoRe)

Inclusion Process with absorbing reservoirs j i SIP L         abs L       L abs 1 , 0 f 2 f f N 1 1 Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i Cristian Giardin` a (UniMoRe)

j i                 2 2 2 2 E x x T P T P T T P P i j L R L R Cristian Giardin` a (UniMoRe)

Energy covariance 2 i ( N + 1 − j ) � � � � � � ( N + 3 )( N + 1 ) 2 ( T R − T L ) 2 ≥ 0 x 2 i x 2 x 2 x 2 − E = E E j i j Remark : up to a sign, covariance is the same in the boundary driven Exclusion Process. Cristian Giardin` a (UniMoRe)

A larger picture & redistribution models (i). Brownian Energy Process BEP ( m ) (ii). Instantaneous thermalization (iii). Symmetric exclusion (SEP(n)) Cristian Giardin` a (UniMoRe)

(i) Brownian Energy Process: BEP The energies of the Brownian Momentum Process z i ( t ) = x 2 i ( t ) Cristian Giardin` a (UniMoRe)

(i) Brownian Energy Process: BEP The energies of the Brownian Momentum Process z i ( t ) = x 2 i ( t ) evolve with Generator � ∂ � ∂ � 2 � − ∂ − 1 − ∂ L BEP = � z i z j 2 ( z i − z j ) ∂ z i ∂ z j ∂ z i ∂ z j ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

Generalized Brownian Energy Process: BEP(m) m � 2 � ∂ ∂ L BMP ( m ) = � � x i ,α − x j ,β ∂ x j ,β ∂ x i ,α α,β = 1 ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

Generalized Brownian Energy Process: BEP(m) m � 2 � ∂ ∂ L BMP ( m ) = � � x i ,α − x j ,β ∂ x j ,β ∂ x i ,α α,β = 1 ( i , j ) ∈ E z i ( t ) = � m α = 1 x 2 The energies i ,α ( t ) evolve with Generator � ∂ � ∂ � 2 � − ∂ − m − ∂ L BEP ( m ) = � z i z j 2 ( z i − z j ) ∂ z i ∂ z j ∂ z i ∂ z j ( i , j ) ∈ E Stationary measures: product Gamma ( m 2 , θ ) Cristian Giardin` a (UniMoRe)

Adding-up SU ( 1 , 1 ) spins j + m 2 � � L ( m ) = � K + i K − j + K − i K + j − 2 K o i K o 8 ( i , j ) ∈ E K + i , K − i , K o � � satisfy SU ( 1 , 1 ) i i ∈ V Cristian Giardin` a (UniMoRe)

Adding-up SU ( 1 , 1 ) spins j + m 2 � � L ( m ) = � K + i K − j + K − i K + j − 2 K o i K o 8 ( i , j ) ∈ E K + i , K − i , K o � � satisfy SU ( 1 , 1 ) i i ∈ V K + K + ξ i + m   � � i = z i i | ξ i � = | ξ i + 1 � 2           K − i = z i ∂ 2 z i + m K − 2 ∂ z i i | ξ i � = ξ i | ξ i − 1 �        i = z i ∂ z i + m  K o K 0 i | ξ i � = ( ξ i + m ) | ξ i �   4 Cristian Giardin` a (UniMoRe)

Generalized Symmetric Inclusion Process: SIP(m) Generator L SIP ( m ) f ( ξ ) = ξ j + m ξ i + m � � � � � [ f ( ξ i , j ) − f ( ξ )] + ξ j [ f ( ξ j , i ) − f ( ξ )] ξ i 2 2 ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

Duality between BEP(m) and SIP(m) Theorem 4 The process { z ( t ) } t ≥ 0 with generator L BEP ( m ) and the process { ξ ( t ) } t ≥ 0 with generator L SIP ( m ) are dual on Γ( m 2 ) � z ξ i D ( z , ξ ) = Γ( m i 2 + ξ i ) i ∈ V Cristian Giardin` a (UniMoRe)

(ii) Redistribution models Generator L KMP f ( z ) = � 1 � dp [ f ( z 1 , . . . , p ( z i + z i + 1 ) , ( 1 − p )( z i + z i + 1 ) , . . . , z N ) − f ( z )] 0 i KMP model is an instantaneous thermalization limit of BEP(2). Cristian Giardin` a (UniMoRe)

Instantaneous thermalization limit L IT i , j f ( z i , z j ) Cristian Giardin` a (UniMoRe)

Instantaneous thermalization limit � � e tL BEP ( m ) L IT i , j f ( z i , z j ) := lim − 1 f ( z i , z j ) i , j t →∞ Cristian Giardin` a (UniMoRe)

Instantaneous thermalization limit � � e tL BEP ( m ) L IT i , j f ( z i , z j ) := lim − 1 f ( z i , z j ) i , j t →∞ � dz ′ i dz ′ j ρ ( m ) ( z ′ i , z ′ j | z ′ i + z ′ j = z i + z j )[ f ( z ′ i , z ′ = j ) − f ( z i , z j )] Cristian Giardin` a (UniMoRe)

Instantaneous thermalization limit � � e tL BEP ( m ) L IT i , j f ( z i , z j ) := lim − 1 f ( z i , z j ) i , j t →∞ � dz ′ i dz ′ j ρ ( m ) ( z ′ i , z ′ j | z ′ i + z ′ j = z i + z j )[ f ( z ′ i , z ′ = j ) − f ( z i , z j )] � 1 dp ν ( m ) ( p ) [ f ( p ( z i + z j ) , ( 1 − p )( z i + z j )) − f ( z i , z j )] = 0 Cristian Giardin` a (UniMoRe)

Instantaneous thermalization limit � � e tL BEP ( m ) L IT i , j f ( z i , z j ) := lim − 1 f ( z i , z j ) i , j t →∞ � dz ′ i dz ′ j ρ ( m ) ( z ′ i , z ′ j | z ′ i + z ′ j = z i + z j )[ f ( z ′ i , z ′ = j ) − f ( z i , z j )] � 1 dp ν ( m ) ( p ) [ f ( p ( z i + z j ) , ( 1 − p )( z i + z j )) − f ( z i , z j )] = 0 � m X � m 2 , m � � X , Y ∼ Gamma 2 , θ i . i . d . = ⇒ P = X + Y ∼ Beta 2 For m = 2: uniform redistribution Cristian Giardin` a (UniMoRe)

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = ( ξ 1 , . . . , ξ | V | ) ∈ { 0 , 1 , 2 , . . . , n } | V | Cristian Giardin` a (UniMoRe)

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = ( ξ 1 , . . . , ξ | V | ) ∈ { 0 , 1 , 2 , . . . , n } | V | L SEP ( n ) f ( ξ ) = � ξ i ( n − ξ j )[ f ( ξ i , j ) − f ( ξ )] + ( n − ξ i ) ξ j [ f ( ξ j , i ) − f ( ξ )] ( i , j ) ∈ E Cristian Giardin` a (UniMoRe)

(iii) Generalized Symmetric Exclusion Process, SEP(n) [Sch¨ utz] Configuration ξ = ( ξ 1 , . . . , ξ | V | ) ∈ { 0 , 1 , 2 , . . . , n } | V | L SEP ( n ) f ( ξ ) = � ξ i ( n − ξ j )[ f ( ξ i , j ) − f ( ξ )] + ( n − ξ i ) ξ j [ f ( ξ j , i ) − f ( ξ )] ( i , j ) ∈ E Stationary measures: product with marginals Binomial(n,p) Cristian Giardin` a (UniMoRe)